Fixed Point of the Finite System DMRG

The density matrix renormalization group (DMRG) is a numerical method that optimizes a variational state expressed by a tensor product. We show that the ground state is not fully optimized as far as we use the standard finite system algorithm, that uses the block structure B • •B. This is because the tensors are not improved directly. We overcome this problem by using the simpler block structure B • B for the final several sweeps in the finite iteration process. It is possible to increase the numerical precision of the finite system algorithm without increasing the computational effort. Establishment of the density matrix renormalization group (DMRG) by White [1] is one of the major progresses in computational condensed matter physics. DMRG enables us to calculate ground states of relatively large scale one-dimensional (1D) quantum systems. [2, 3, 4, 5, 6]. Two-dimensional (2D) classical systems, [7, 8, 9, 10] and 1D quantum system at finite temperature [11, 12, 13, 14] have also been investigated. Östlund and Rommer [15] examined the thermodynamic limit (N → ∞) of the infinite system algorithm, and they pointed out that the block state B corresponds to a product of position independent tensor. It should be noted that their result does not show that the infinite system algorithm creates a translationally invariant — position independent — variational state for the whole system B••B, where “•” denotes a bare spin variable between the left and the right blocks. Actually, the variational state has a slight position dependence. For example, the bond energy 〈••〉 at the center of the antiferromagnetic S = 1/2 Heisenberg chain, which is calculated by the infinite system algorithm, is lower than the exact ground state energy per site. [16] Such a position dependence in the variational state spoils the numerical efficiency of the infinite system algorithm. [17] As we show in the following, the finite system algorithm does not fully improve the variational state in the same reason. The purpose of this letter is to remove the source of such a numerical error, and to increase the numerical precision in DMRG. Let us briefly review the construction of the variational state, which is used in the standard finite system algorithm. We consider the IRF model [18] as a reference system, whose transfer matrix is written as the product of local Boltzmann weights T s′1···s ′ N s 1··· sN = W s′1s ′